# Lid-Driven Cavity in 2D

## Problem Description

A square cavity with a side length of L°= 1e-3° m is filled with water (density: 1000 kg/m3, dynamic viscosity: 1e-3 kg/m/s). The upper wall is moved with a constant velocity U0 imposing a rotational movement of the fluid in the cavity. Here, the configurations with Re°=°100, Re°=°1000 and Re°=°10000 are simulated by adjusting the velocity of the lid correspondingly. The highly-resolved Finite-Difference calculations by Ghia et al with a grid resolution of 257x257 are taken as reference.

## Numerical Setup

Three different resolutions of 50x50, 100x100, and 200x200 fluid particles are used for each Reynolds-Number. Figure 1 shows the initial particle distribution for the coarsest resolution where wall boundary conditions are imposed by three rows of particles. The simulation is run until a fully developed flow is achieved.

## Results

The velocity profiles in the horizontal and vertical centerlines are shown in Figure 2 to Figure 4 for each Reynolds-Number. For the Re = 100 case, all the profiles show very good agreement with the reference profiles by Ghia et al. Velocity profiles at Re = 1000 and Re = 10000 match those of Ghia et al at finer resolutions of 100x100 and 200x200, respectively. This is expected as higher Reynolds numbers results in finer structures and requires more particles to resolve.

S. Adami, H. Hu and N. Adams, "A generalized wall boundary condition for smoothed particle hydrodynamics," Journal of Computational Physics, vol. 231, pp. 7057-7075, 2012.

U. Ghia and K. S. C. Ghia, "High-Re solutions for incompressible flow using the Naview-Stokes equations and a multigrid method," Journal of Computational Physics, vol. 48, pp. 387-411, 1982.